Finally, to assess the predictive performance of our ensembles, we compute the CRPS and report the average, over all three state components and 2000 assimilation cycles, for increasing \(M\)  in Figure REF  (left). For \(M \ge 60\)  samples, the nonlinear stochastic map filter yields lower values of CRPS, which provides a simultaneous diagnostic of calibration and sharpness as defined in [1]}. With increasing nonlinearity in the prior-to-posterior transformation, the average CRPS for the Lorenz-63 model approaches that of a particle filter using \(10^{6}\)  samples.
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